On the Approximation of Submodular Functions

On the Approximation of Submodular Functions

Nikhil R. Devanur111Microsoft Research, Redmond    @microsoft.com    Shaddin Dughmi222University of Southern California   shaddin@usc.edu    Roy Schwartz    Ankit Sharma333Carnegie Mellon University   ankits@cs.cmu.edu    Mohit Singh
Abstract

Submodular functions are a fundamental object of study in combinatorial optimization, economics, machine learning, etc. and exhibit a rich combinatorial structure. Many subclasses of submodular functions have also been well studied and these subclasses widely vary in their complexity. Our motivation is to understand the relative complexity of these classes of functions. Towards this, we consider the question of how well can one class of submodular functions be approximated by another (simpler) class of submodular functions. Such approximations naturally allow algorithms designed for the simpler class to be applied to the bigger class of functions. We prove both upper and lower bounds on such approximations. Our main results are:

  • General submodular functions444We additionally assume that the submodular function takes value 0 on the null set and the universe. can be approximated by cut functions of directed graphs to a factor of , which is tight.

  • General symmetric submodular functions can be approximated by cut functions of undirected graphs to a factor of , which is tight up to a constant.

  • Budgeted additive functions can be approximated by coverage functions to a factor of , which is tight.

Here is the size of the ground set on which the submodular function is defined. We also observe that prior works imply that monotone submodular functions can be approximated by coverage functions with a factor between and .

1 Introduction

Submodular optimization problems have been a rich area of research in recent years, motivated by the principle of diminishing marginal returns which is prevalent in real world applications. Such functions are ubiquitous in diverse disciplines, including economics, algorithmic game theory, machine learning, combinatorial optimization and combinatorics. While submodular function can be minimized efficiently, i.e., in polynomial time [16, 23, 18], many natural optimization problems over submodular functions are NP-hard, e.g., Max--Coverage [20], Max-Cut and Max-DiCut [14], and Max-Facility-Location [9]. Consequently, many works, specifically in the setting of algorithmic game theory [7, 10, 11, 17], have explored simpler subclasses of submodular functions for which the given algorithmic problem can still be well-approximated. Such subclasses of submodular functions have included cut functions of graphs, coverage functions of set systems, budgeted additive functions, matroid rank functions, etc.

Our work is motivated by the question, how complex can a submodular function be? Since this is such a fundamental question, it has been asked in different forms previously. Goemans et al. [13] consider how many queries to a submodular function are sufficient to infer the value of the function, approximately, at every point in the domain. Balcan and Harvey [2] focus on the problem of learning submodular functions in a probabilistic model; are few random queries enough to infer the value at almost all points in the domain? Badanidiyuru et al. [1] ask whether an approximate sketch of a submodular function, or more generally a subadditive function, exists (i.e., can the function be represented in polynomial space)? Seshadri and Vondrák [24] consider the testability of submodular functions: how many queries does it take to check whether a function is close to being submodular?

We approach this question by noting that not all submodular functions are identically complex and some have been more amenable to optimization than others. Thus, one natural way to characterize the relative complexity of one class of submodular functions w.r.t another, is to ask how well can a function in the first class be approximated by a function in the second. Formally, we ask the following question. Given two classes of submodular functions and (typically ), what is the smallest , such that for every , there exists a such that for each ? Here class would represent the class of submodular functions which are easier to optimize for some problem and class would represent a bigger class which we want to optimize over. We also note that this concept of approximation is not special to submodular functions and can be asked for any two classes of functions. We focus on submodular functions due to their ubiquitous nature in optimization.

Intuitively, this notion of approximation resembles the long and rich line of work that deals with the algorithmic applications of geometric embeddings, in which the goal is to embed hard metric spaces into simpler ones. Some successful examples include embedding general metrics into normed spaces [5], dimension reduction in a Euclidean space [19] and the probabilistic embedding into ultrametrics [4, 12]. As in the metric case, a natural byproduct of the above approach is that if there exists an -approximation algorithm for any submodular function in , then there exists a -approximation algorithm for all functions in . As an application of our approach, we show how to obtain an algorithm for the online submodular function maximization problem, for general monotone submodular functions [6]. Previously, results were known for only certain subclasses of submodular functions; see Appendix E for details.

1.1 Our Results and Techniques

We start by asking how well a general submodular function (with the additional property that ) can be approximated by a function in the canonical simpler subfamily of non-symmetric submodular functions, cut function of a directed graph. We give matching upper and lower bounds for such an approximation (Theorem 1.1). Next, we ask the same question for symmetric submodular functions vis-a-vis its canonical simpler subfamily, cut functions of undirected graphs. In this case, we provide nearly matching upper and lower bounds (Theorem 1.2). We then move our attention to two subfamilies, budgeted additive functions and coverage functions, both of which, as already mentioned in the introduction, have received considerable interest in the algorithmic game theory setting. We show tight upper and lower bounds for approximating budgeted additive functions with coverage functions (Theorem 1.3). These results are summarized in Table 1. While previous works [13, 2, 1] studied the complexity of submodular functions from different perspectives, they do imply some additional results, both positive and negative, on the approximation of monotone submodular functions by simpler classes of submodular functions (as illustrated in Table 1 and discussed in detail in Appendix G).

Input Class Output Class Approximation Approximation
Upper Bound Lower Bound
General Submodular Cut Functions (directed)
Symmetric Submodular Cut functions (undirected)
Budgeted Additive Coverage
Monotone Submodular Coverage/Budgeted Additive  [13]  [2, 1]
Table 1: Our results are described in the first three rows. The results in the last row are either implicit in the references or follow as a corollary (Appendix G). When the output class is a cut function of a graph, we assume that the input function satisfies , as every cut function must satisfy this constraint. Here, denotes the size of the ground set.

Let us now briefly discuss the main techniques that we use to obtain our results. In contrast to previous works [13, 2, 1], arbitrary submodular functions, as opposed to monotone submodular functions, present different challenges. As an illustration, for approximating a submodular function via a cut function of a graph , consider the case when there is a non-trivial set for which . Then the weight of the cut in is forced to be zero. Indeed, all sets with must be in a correspondence with cuts in of value zero. Thus, given a submodular function , our construction of optimizes for the minimizers of the submodular function . Surprisingly, this can be shown to give the best possible approximation. For a symmetric submodular function , we show that it suffices to use the cut function of a tree (as opposed to a general undirected graph) utilizing the Gomory-Hu tree representation [15, 21] of .

For approximating budgeted additive functions by coverage functions, we first give a randomized construction achieving an approximation factor of . We then show that this is the best possible approximation factor as characterized by a linear program. The proof of the lower bound of uses linear programming duality and proceeds by presenting a feasible dual solution to the linear program achieving an objective value of in the limit. We would like to point out that all our results are algorithmic and the claimed approximations can be found in polynomial time given a value oracle for the submodular function.

1.2 Related Work

Goemans et al. [13] considered the problem of how well a given monotone submodular function can be approximated when only polynomially many value oracle queries are permitted. They presented an approximation of , and an improved guarantee of in the case that is a matroid rank function. Implicit in this algorithm and relevant to our setting is an approximation of all monotone submodular functions by budgeted additive functions (Appendix G). The current best lower bound for the problem studied by [13] is given by Svitkina and Fleischer [25] and is .

Balcan and Harvey [2] take the learning perspective to the study of the complexity of submodular functions. They study the problem of probabilistically learning a monotone submodular function, given the values the function takes on a polynomial sized sample of its domain. They provide a lower bound of on the best possible approximation a learning algorithm can give to the submodular function, even when it knows the underlying sampling distribution, and the submodular function to be learned is Lipschitz. Another result with this perspective is by Balcan et al. [3] who show that a symmetric non-monotone submodular function can be approximated to within by the square root of a quadratic function. Furthermore, they show how to learn such submodular functions.

Badanidiyuru et al. [1], motivated by the problem of communicating bidders’ valuations in combinatorial auctions, study how well specific classes of set functions can be approximated given the constraint that the approximating function be representable in polynomially many bits. They named such an approximation a sketch, proving that coverage functions admit sketches with an arbitrarily good approximation factors. Additionally, for the larger class of monotone subadditive functions, they construct sketches that achieve an approximation of . Combining the results of Badanidiyuru et al. [1] and Balcan and Harvey [2], a lower bound of follows for the approximation of monotone submodular functions by budgeted additive functions (Appendix G).

Testing of submodular functions has been studied recently by Seshadri and Vondrák [24] for general monotone submodular functions and for coverage functions by Chakrabarty and Huang [8]. The goal here is to query the function on few domain points and answer whether a function is close to being submodular or not. The measure of closeness is the fraction of the domain in which the function needs to be modified so as to make it submodular.

1.3 Preliminaries and Formal Statement of Results

Given a ground set , a function is called submodular if for all subsets , we have . A submodular function is called non-negative if for each . In this paper, we only consider non-negative submodular functions. A submodular function is called symmetric if for each , and monotone if for each . We say that a class of functions -approximates another class , if for every there exists a such that for any . We denote by the size of the ground set . We now define certain subclasses of submodular functions that we consider in the paper.

Definition 1 (Coverage function).

A function is a coverage function if there exists an auxiliary ground set , a weight function and family of subsets such that , .

Definition 2 (Budgeted additive function).

A function is a budgeted additive function if there exist non-negative reals for each , and a non-negative real such that , .

It is well known that coverage functions and budgeted additive functions are monotone submodular functions.

Definition 3 (Cut function).

A function is a directed cut function if there exists a directed graph with non-negative arc weights , such that , , where denotes the set of outgoing arcs, with their tails in and heads in , and for any subset of arcs.

Similarly, one can define to be the undirected cut function of an undirected graph by substituting with , the set of edges with exactly one endpoint in . It is well known that cut functions, whether directed or undirected, are submodular. Furthermore, clearly, undirected cut functions are symmetric.

Let us now formally state our main results:

Theorem 1.1.

Let be a non-negative submodular function with . Then the class of directed cut functions -approximates . Moreover, there exists a non-negative submodular function with such that any directed cut function cannot approximate within a factor better than .

Theorem 1.2.

Let be a non-negative symmetric submodular function with . Then the class of undirected cut functions -approximates . Moreover, there exists a symmetric submodular function with such that any undirected cut function cannot approximate within a factor better than .

Theorem 1.3.

Let be a budgeted additive function. Then coverage functions -approximates . Moreover, for every fixed , there exists a budgeted additive function such that any coverage function cannot approximate within a factor better than .

Theorems 1.1, 1.2 and 1.3, are proved in Sections 2,  3 and 4 respectively.

2 Approximating General Submodular Functions by Directed Cut Functions of Graphs

In this section we prove Theorem 1.1 which provides a tight approximation of a non-negative submodular function using a directed cut function of a graph . Before proving the main result of this section, we first state a technical lemma, whose proof we defer to Appendix C.

Lemma 2.1.

For every submodular function , and any collection of sets , , , : .

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1.

Upper Bound: Given a submodular function , we construct a directed graph with non-negative weights on the arcs such for every , . For every and , introduce a directed arc from to with weight: where . We start by proving that:

(1)

If or , then clearly and (1) holds. We now restrict our attention to the case where . For any note that , since the definition of implies that for all . Additionally, for any note that , since the definition of implies that . Thus, one can conclude that and therefore,

Inequality (i) is derived from the fact that is submodular and non-negative. Inequality (ii) is derived from Lemma 2.1 and the definition of . This concludes the proof of (1).

We continue by proving that:

(2)

If or , then clearly and (2) holds. We now restrict our attention to the case where . Note that for any and , by the definition of : . Thus, one can conclude that:

Equality (i) is by the definition of weights . Inequality (ii) is be the definition of . Inequality (iii) is by the fact that the number of pairs is at most . This concludes the proof of (2). Combining both (1) and (2) concludes the proof of the upper bound of the theorem.

Lower Bound: Assume that is even and fix an arbitrary of size . Consider the following function

Namely, is the indicator function that hits but does not hit all of . A simple check shows that is submodular. Let be a weighted graph with non-negative weights on the arcs whose directed cut function satisfies for each set , for some . We will show that proving the lower bound.

First, we prove that the arcs with non-zero weight must go from to . We consider the following cases.

  1. Consider an edge . But and since . Thus, .

  2. Consider an edge . But and since . Thus, .

  3. Consider an edge . But and , since . Hence, .

Therefore, all arcs with non-zero weight must go from to . For any and note that since:

(3)

Inequality (i) is derived from the fact for each set . Equality (ii) is by the definition of . Furthermore, note that:

(4)

Inequality (i) is derived from inequality (3). Inequality (ii) is derived from the fact that for each set . Equality (iii) is by the definition of . Note that inequality (4) implies that , thus, concluding the proof of the lower bound of the theorem. ∎

3 Approximating Symmetric Submodular Functions by Undirected Cut Functions of Graphs

In this section, we prove Theorem 1.2 which provides upper and lower bounds on the approximation of a symmetric submodular function using an undirected cut function of a graph. For the upper bound, our algorithm uses Gomory-Hu trees of symmetric submodular functions [15, 21]. Given a symmetric non-negative submodular function , a tree is a Gomory-Hu tree if for every edge : , where is one of the two connected components obtained after removing from (since is symmetric, it does not matter which one of the two connected components we choose). In other words, in a Gomory-Hu tree, the cut induced in , corresponds to a minimum value subset that separates and . We prove that the cut function of the Gomory-Hu tree of is a good approximation.

Proof of Theorem 1.2.

Upper Bound: Let be a symmetric submodular function. We shall construct an undirected tree with non-negative weights on the edges such that for every , . We set to be a Gomory-Hu tree of and let the weight of any edge to be where is the one of the two connected components obtained after removing edge . As mentioned above, the weight of edge is the minimum of over all separating and .

Fix an arbitrary and denote by all the edges crossing the cut that defines in . Let denote the partition of induced by deleting the edges from . Furthermore, denote by the non-empty sets in . Observe that is a partition of . Since each , , has exactly one vertex in and the other in , we can associate with a unique set from , the set containing one of the endpoints of . Additionally, let us denote to be the edges which are associated with set for each . Clearly form a partition of .

We claim that for every :

(5)

Recall that is one of the connected component after removing edge from . Since is a subset of a connected component formed after removing all the edges from , it must be contained in one of the components formed after removing edge from . Without loss of generality, we assume that for each edge . It is straightforward to see that . Now, we have

Inequality (i) is derived from the fact that is submodular and non-negative. Equality (ii) is derived from the symmetry of . Equality (iii) is derived from the fact that .

We start by proving that:

(6)

This can be proved as follows:

Equality (i) is by the definition of edge weights in . Equality (ii) is by the fact that form a partition of . Inequality (iii) is derived from inequality (5). Inequality (iv) is derived from the fact that is submodular and non-negative. This concludes the proof of (6).

We continue by proving that:

(7)

Let and be the endpoints of edge and without loss of generality assume that and . Note that for every , since is a candidate set separating and . Hence, one can conclude that:

Equality (i) is by the definition of edge weights in . Inequality (ii) is what we proved above, and inequality (iii) is derived from the fact that contains at most edges, thus, . This concludes the proof of (7). Combining both (6) and (7) concludes the proof of the upper bound of the theorem.

Lower Bound: Refer to Appendix A. ∎

4 Approximating Budgeted Additive Functions by Coverage Functions

In this section, we present matching upper and lower bounds for approximating budgeted additive functions by coverage functions (Theorem 1.3)555It is easy to show that a coverage function can be written exactly as a sum of budgeted additive functions.. The following lemma from Chakrabarty and Huang [8] provides the alternate representation of coverage functions used in our proof of the lower bound.

Lemma 4.1.

[8] A function is a coverage function if and only if there exist reals for each such that for each .

Proof of Theorem 1.3.

Upper Bound: Refer to Appendix B.
Lower Bound: We will construct a budgeted additive function which cannot be approximated by coverage functions to factor better than for any . We will consider the family of budgeted additive function , parameterized by the size of domain they are defined on, where for some integer . Under , all items have value one and the budget is . Please note that these also constitute a family of uniform matroid rank functions. Therefore,

(8)

Let be a coverage function that gives the maximum value of such that and be the value of as given by . Observe that here function is always smaller than the function . The function would give a -approximation for approximating function . This slight change in notation helps for exposition below. We shall show that as , tends to a value that is at most . This shall prove our claim.

Using Lemma 4.1, we note that can be characterized by a solution to a linear problem given below. Here, the variables are , one for each set . The dual (D) of this linear program is given alongside. We will construct a dual solution of value approaching as . Since every feasible dual solution is an upper bound on , the result follows.

Since, is symmetric across sets of the same cardinality, we can assume, without loss of generality, that the optimal dual solution is also symmetric. Specifically, the values of the dual variables and shall depend only the cardinality . Let us write the symmetrized dual program.

Symmetrized Dual Program
subject to
(9)
(10)

Let denote the coefficient of in the equation corresponding to set size , i.e., . Further, define .

We give the following solution to the dual linear program. Let , and . Rest of the variables are set to zero.

We first show that the above solution is feasible for the dual and has objective value that tends to as . It is easy to see that with the proposed setting of , Equation (10) is satisfied. To show that Equation (9) is satisfied, we show that , . Using our notation, it suffices to show that for all , .

Claim 4.2.

is an increasing function of and is a decreasing function of .

Proof.

. ∎

Claim 4.3.

For all , .

Proof.
  • For such that :

  • For with :

where the second inequality in both the cases follows because is a decreasing function in . ∎

Let us now bound the value of the dual objective function. Look at the value that the dual objective function attains with this setting of variables. The function value is . Since , the dual objective value is equal to

This quantity tends to as . ∎

5 Future Directions

We mention here a couple of main research questions that are left open by the present work. The first is how well a non-negative monotone submodular function can be approximated by the sum of matroid rank functions. Dughmi et al. [11] show that the Hessian matrix for a matroid rank sum has to be negative semi-definite, and it is easy to come with a budgeted additive function that does not obey this property. Hence, we cannot hope for the best approximation factor for a submodular function by a matroid rank sum to be 1; in fact we can show that the approximation factor cannot be better than some constant bounded away from 1. In terms of positive results, a factor approximation follows from [13] and a follows from a result in Section 44.6(B) in [22].

The second is approximating a non-negative symmetric submodular function by a hypergraph cut function (in this paper, we only considered graph cut functions). The lower bound example in the paper for graph cut functions can be extended to show that a -regular hypergraph cannot approximate to a factor better than . In terms of positive results, we know no better than the ones mentioned in this paper.

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Appendix A Lower Bound of Theorem 1.2

Lower Bound of Theorem 1.2.

Consider the following symmetric submodular function :

Let be an edge weighted graph with non-negative weights on the edges whose cut function satisfies for each set for some . We will show that .

For any vertex , . Thus, the total weight of edges in is at least . Every undirected graph has a non-trivial cut that contains at least half the total weight of edges in the graph, thus, there exists a cut , , where . The existence of such a cut can be shown by picking a cut at random where each vertex is in with probability independently. The expected weight of the cut will be exactly half the total weight of all edges. Now, we have , concluding the proof of the lower bound of the theorem. ∎

Appendix B Upper Bound of Theorem 1.3

Upper Bound of Theorem 1.3.

Consider any budgeted additive function over some domain , with budget and the values of the elements be denoted by where . Without loss of generality, we assume all these values to be integers. Take an auxiliary ground set of size . For each , construct a set , formed by choosing points (with replacement) at random from . Consider function , defined as for all .

By definition, is a coverage function. Furthermore, it is easy to see that for all , . We now show that , where the expectation is taken over the randomness of the procedure described to construct . Note that is a coverage function. Consider any set . Let , i.e., . Consider the case when . Consider any point in auxiliary ground set . The probability that this point is not covered by any of the sets for is at most . Hence, the expected value of is at least . Here we use the inequality . Hence, . The proof for the case when is similar. Thus for each set , we have

Thus, we obtain the function approximates within a factor of . ∎

Appendix C Proof of Lemma 2.1

Proof of Lemma 2.1.

The following inequalities are derived from the definition of submodularity:

Summing up the above inequalities and canceling common terms on the two sides and using the fact that is non-negative we obtain that . ∎

Appendix D Uniform Submodular and Matroid Rank Functions

Definition 4 (Uniform Submodular Function).

A submodular function is said to be uniform if the value it takes on a set depends only on the cardinality of the set.

Lemma D.1.

Any non-negative, integer-valued, monotone, uniform, submodular function is -approximated by a sum of uniform matroid rank functions, and hence by a sum of budgeted additive functions.

Proof.

Consider an integer-valued, non-negative, monotone, uniform, submodular function over the universe and let be the value takes for sets of cardinality . Consider uniform matroid rank functions where

(11)

We claim that there exist a set of ’s, such that for all such that

(12)

It is easy to see that the above claim implies that for all .

Now, we prove the claim. If, for every , we substract equation from , we get the following set of equations

(13)

From here, we can see that the following assignments to is a valid solution to the above equations. Set and for , set . All the ’s are positive since is monotone and is submodular.

Finally, it is easy to see that every uniform matroid rank function is also a budgeted additive function with all elements having value one, and the budget equal to the rank of the matroid. ∎

Appendix E Application to Online Submodular Function Maximization

We consider the problem of online submodular function maximization as studied by Buchbinder et. al [6]. We are given a universe and matroid . In an online manner, at each step for , we are given a monotone submodular function . The goal is to maintain an independent set at any step such that . The objective value to maximize is . As in the notion of competitive analysis, any algorithm is compared to the best offline optimum .

Buchbinder et. al [6] give a -competitive algorithm when each of the submodular function is weighted matroid rank function where . In particular, the result applies when each of the functions is a coverage function.

Using the fact every monotone submodular function can be approximated by a coverage function to a factor of , we directly obtain the following corollary.

Corollary E.1.

There is a -competitive online algorithm for the online submodular function maximization problem when each of the submodular functions is an arbitrary monotone submodular function.

Appendix F Approximating Monotone Submodular Functions by Coverage Functions and by Budgeted Additive Functions

The two main results of the section are the following.

Theorem F.1.

Coverage functions can approximate every non-negative monotone submodular function to within a factor . Additionally, the class of coverage functions cannot approximate every non-negative monotone submodular function to a factor within .

Theorem F.2.

The class of sum of budgeted additive functions can approximate every non-negative monotone submodular function to a factor within . Additionally, the class of sum of budgeted additive functions cannot approximate every non-negative monotone submodular function to a factor within .

f.1 Upper Bound

For the upper-bound, we show that budgeted additive functions can -approximate the class of non-negative, monotone, submodular functions. Then we use Theorem 1.3 to infer that the coverage functions too can give approximately the same guarantee.

Lemma F.3.

The class of sum of budgeted additive functions can approximate every non-negative monotone submodular function to within a factor .

The following corollary follows easily from Lemma G.3 and Theorem 1.3

Corollary F.4.

The class of sum of coverage functions can approximate every non-negative monotone submodular function to factor .

The proof of Lemma G.3 follows from Lemmas G.5 and G.6. Lemma G.5 [13] gives a particular function that -approximates a general monotone, non-negative, sub-modular function, and Lemma G.6 implies that this approximating function can be written as a sum of budgeted additive functions.

Lemma F.5.

[13] For every monotone submodular function , there exists positive reals for each such that defined as , approximates within factor .

Lemma F.6.

Every submodular function of the form , where and is a non-negative, monotone, concave and integer valued on integral inputs, can be written as a sum of budgeted additive functions.

Proof.

Let . Consider the function , over the domain , defined as for all . Since is a non-negative, monotone, concave function, it is easy to verify that is a non-negative, monotone, submodular function. Construct mutually disjoint sets such that . Clearly, .

From Lemma D.1, we know that can be expressed as where each is a uniform matroid rank function with rank , over the domain and each .

This implies that for all , . Now, for every , construct the budget additive function , defined as , where for , the value and the budget is . Since is a uniform matroid rank function of rank and ’s are mutually disjoint, we have for all ,

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Therefore, we get for all sets , . ∎

f.2 Lower Bound

For the lower bound, we first show that sum of coverage functions cannot approximate the class of monotone, submodular functions well, and then use Theorem 1.3, to infer that, therefore, even the class of sum of budgeted additive functions cannot approximate a monotone submodular function well.

Lemma F.7.

The class of sum of coverage functions cannot approximate every non-negative monotone submodular function to a factor within .

An easy corollary of Lemma G.7 that follows from Theorem 1.3 is the following.

Corollary F.8.

The class of sum of budgeted additive functions cannot approximate every non-negative monotone submodular function to a factor within .

We now present the proof of Lemma G.7. We will need to use results from [1] and [2], for which we first present a definition.

Definition 5.

A -sketch of a function is a polynomially sized (in and ) representable function such that , .

The following result is from [1].

Lemma F.9.

[1] Coverage functions allow from arbitrary well sketches i.e., for any , there exists a sketch.

The following result is from [2]. It gives a ‘large’ family of matroid rank functions, such that any two functions in the class have at least one point where the values that they take differ by a ‘significant’ factor.

Lemma F.10.

[2] For any , there exists a family of sets with and a family of matroids such that for all , it is the case that

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where is the rank function of the matroid

of Lemma g.7.

Let the class of matroid rank functions on the domain of size be -approximable by coverage functions, for some . That is, for a domain , for all matroid rank function , there exists a coverage function such that .

By Lemma G.9, for every and every coverage function , there exists a polynomially sized (polynomial in and ) representable function such that . Hence, for all and for all matroid rank functions , there exists a polynomial sized representable function such that . For any given , there are only many different functions.

From Lemma G.10, for , there exists family of sets with